A Numerical Criterion for Simultaneous Normalization

We investigate conditions for simultaneous normalizability of a family of reduced schemes, i.e., the normalization of the total space normalizes, fiber by fiber, each member of the family. The main result (under more general conditions) is that a flat family of reduced equidimensional projective C-varieties (Xy)y∈Y with normal parameter space Y—algebraic or analytic—admits a simultaneous normalization if and only if the Hilbert polynomial of the integral closure OXy is locally independent of y. When the Xy are curves projectivity is not needed, and the statement reduces to the well known δ-constant criterion of Teissier. Proofs are basically algebraic, analytic results being related via standard techniques (Stein compacta, etc.) to more abstract algebraic ones.